3.27.76 \(\int \frac {8+(3+8 x) \log (-12-32 x)}{(3+8 x) \log (-12-32 x)} \, dx\)

Optimal. Leaf size=13 \[ x+\log \left (\log \left (16 \left (-\frac {3}{4}-2 x\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6688, 2390, 12, 2302, 29} \begin {gather*} x+\log (\log (-4 (8 x+3))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + (3 + 8*x)*Log[-12 - 32*x])/((3 + 8*x)*Log[-12 - 32*x]),x]

[Out]

x + Log[Log[-4*(3 + 8*x)]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {8}{(3+8 x) \log (-12-32 x)}\right ) \, dx\\ &=x+8 \int \frac {1}{(3+8 x) \log (-12-32 x)} \, dx\\ &=x-\frac {1}{4} \operatorname {Subst}\left (\int -\frac {4}{x \log (x)} \, dx,x,-12-32 x\right )\\ &=x+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-12-32 x\right )\\ &=x+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-4 (3+8 x))\right )\\ &=x+\log (\log (-4 (3+8 x)))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 11, normalized size = 0.85 \begin {gather*} x+\log (\log (-4 (3+8 x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + (3 + 8*x)*Log[-12 - 32*x])/((3 + 8*x)*Log[-12 - 32*x]),x]

[Out]

x + Log[Log[-4*(3 + 8*x)]]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 9, normalized size = 0.69 \begin {gather*} x + \log \left (\log \left (-32 \, x - 12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*log(-32*x-12)+8)/(8*x+3)/log(-32*x-12),x, algorithm="fricas")

[Out]

x + log(log(-32*x - 12))

________________________________________________________________________________________

giac [A]  time = 0.37, size = 9, normalized size = 0.69 \begin {gather*} x + \log \left (\log \left (-32 \, x - 12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*log(-32*x-12)+8)/(8*x+3)/log(-32*x-12),x, algorithm="giac")

[Out]

x + log(log(-32*x - 12))

________________________________________________________________________________________

maple [A]  time = 0.40, size = 10, normalized size = 0.77

method	result	size
norman	\(\ln \left (\ln \left (-32 x -12\right )\right )+x\)	\(10\)
risch	\(\ln \left (\ln \left (-32 x -12\right )\right )+x\)	\(10\)
derivativedivides	\(x +\frac {3}{8}+\ln \left (\ln \left (-32 x -12\right )\right )\)	\(11\)
default	\(x +\frac {3}{8}+\ln \left (\ln \left (-32 x -12\right )\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x+3)*ln(-32*x-12)+8)/(8*x+3)/ln(-32*x-12),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-32*x-12))+x

________________________________________________________________________________________

maxima [C]  time = 0.90, size = 71, normalized size = 5.46 \begin {gather*} -\frac {3}{8} \, {\left (i \, \pi + 2 \, \log \relax (2) + \log \left (8 \, x + 3\right )\right )} \log \left (i \, \pi + 2 \, \log \relax (2) + \log \left (8 \, x + 3\right )\right ) + \frac {3}{8} \, \log \left (i \, \pi + 2 \, \log \relax (2) + \log \left (8 \, x + 3\right )\right ) \log \left (-32 \, x - 12\right ) + x + \log \left (i \, \pi + 2 \, \log \relax (2) + \log \left (8 \, x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*log(-32*x-12)+8)/(8*x+3)/log(-32*x-12),x, algorithm="maxima")

[Out]

-3/8*(I*pi + 2*log(2) + log(8*x + 3))*log(I*pi + 2*log(2) + log(8*x + 3)) + 3/8*log(I*pi + 2*log(2) + log(8*x
+ 3))*log(-32*x - 12) + x + log(I*pi + 2*log(2) + log(8*x + 3))

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 9, normalized size = 0.69 \begin {gather*} x+\ln \left (\ln \left (-32\,x-12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(- 32*x - 12)*(8*x + 3) + 8)/(log(- 32*x - 12)*(8*x + 3)),x)

[Out]

x + log(log(- 32*x - 12))

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 10, normalized size = 0.77 \begin {gather*} x + \log {\left (\log {\left (- 32 x - 12 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*ln(-32*x-12)+8)/(8*x+3)/ln(-32*x-12),x)

[Out]

x + log(log(-32*x - 12))

________________________________________________________________________________________